Session 2: Options I

C15.0008 Corporate Finance Topics

Summer 2006

Outline

• Call and put options

• The law of one price

• Put-call parity

• Binomial valuation

Options, Options Everywhere!

• Compensation—employee stock options• Investment/hedging—exchange traded and OTC

options on stocks, indexes, bonds, currencies, commodities, etc., exotics

• Embedded options—callable bonds, convertible bonds, convertible preferred stock, mortgage-backed securities

• Equity and debt as options on the firm• Real options—projects as options

Example..

Options

The right, but not the obligation to buy (call) or sell (put) an asset at a fixed price on or before a given date.

Terminology:

Strike/Exercise Price

Expiration Date

American/European

In-/At-/Out-of-the-Money

An Equity Call Option

• Notation: C(S,E,t)

• Definition: the right to purchase one share of stock (S), at the exercise price (E), at or before expiration (t periods to expiration).

Where Do Options Come From?

• Publicly-traded equity options are not issued by the corresponding companies

• An options transaction is simply a transaction between 2 individuals (the buyer, who is long the option, and the writer, who is short the option)

• Exercising the option has no effect on the company (on shares outstanding or cash flow), only on the counterparty

Numerical example

• Call option

• Put option

Option Values at Expiration• At expiration date T, the underlying (stock) has market price ST• A call option with exercise price E has intrinsic value (“payoff to holder”)

• A put option with exercise price E has intrinsic value (“payoff to holder”)

),0max(if0

ifpayoff ES

ES

ESEST

T

TT

),0max(if0

ifpayoff T

T

TT SEES

ESSE

Call Option Payoffs

Payoff

STE

Long CallPayoff

STE

Short Call

Put Option Payoffs

Payoff

STE

Long PutPayoff

STE

Short Put

E

E

Other Relevant Payoffs

Payoff

ST

Stock

Payoff

ST

Risk-Free Zero Coupon BondMaturity T, Face Amount E

E

The Law of One Price

• If 2 securities/portfolios have the same payoff then they must have the same price

• Why? Otherwise it would be possible to make an arbitrage profit– Sell the expensive portfolio, buy the cheap

portfolio– The payoffs in the future cancel, but the

strategy generates a positive cash flow today (a money machine)

Put-Call Parity

Stock + PutPayoff

STE

Payoff

STE

E=

Payoff

STE

Call +Bond

Payoff

STE

E=

Put-Call Parity

Payoffs:

Stock + Put = Call + Bond

Prices:

Stock + Put = Call + Bond

Stock = Call – Put + Bond

S = C – P + PV(E)

Introduction to binomial trees

What is an Option Worth?

Binomial Valuation

Consider a world in which the stock can take on only 2 possible values at the expiration date of the option. In this world, the option payoff will also have 2 possible values. This payoff can be replicated by a portfolio of stock and risk-free bonds. Consequently, the value of the option must be the value of the replicating portfolio.

Payoffs

Stock

100

137

73

Bond (rF=2%)

100

102

102

Call (E=105)

C

32

0

1-year call option, S=100, E=105, rF=2% (annual)1 step per yearCan the call option payoffs be replicated?

Replicating Strategy

Buy ½ share of stock, borrow $35.78 (at the risk-free rate).

Cost(1/2)100 - 35.78 = 14.22

Payoff(½)137 - (1.02) 35.78 = 32

Payoff(½)73 - (1.02) 35.78 = 0

The value of the option is $14.22!

Solving for the Replicating Strategy

The call option is equivalent to a levered position in the stock (i.e., a position in the stock financed by borrowing).

137 H - 1.02 B = 32

73 H - 1.02 B = 0 H (delta) = ½ = (C+ - C-)/(S+ - S-)

B = (S+ H - C+ )/(1+ rF) = 35.78

Note: the value is (apparently) independent of probabilities and preferences!

Multi-Period Replication

Stock

100

80

125

100

156.25

64

Call (E=105)

0

51.25

0

C+

C-

1-year call option, S=100, E=105, rF=1% (semi-annual)2 steps per year

Solving Backwards

• Start at the end of the tree with each 1-step binomial model and solve for the call value 1 period before the end

• Solution: H = 0.911, B = 90.21 C+ = 23.68• C- = 0 (obviously?!)

125

100

156.25

0

51.25 rF = 1%C+

The Answer

• Use these call values to solve the first 1-step binomial model

• Solution: H = 0.526, B = 41.68 C = 10.94• The multi-period replicating strategy has no intermediate

cash flows

100

80

125

0

23.68rF = 1%

Building The Tree

S

S+

S-

S--

S+-

S++ S+ = uS

S- = dS

S++ = uuS

S-- = ddS

S+- = S-+ = duS = S

The Tree!

u =1.25, d = 0.8

100

80

125

100

156.25

64

Binomial Replication

• The idea of binomial valuation via replication is incredibly general.

• If you can write down a binomial asset value tree, then any (derivative) asset whose payoffs can be written on this tree can be valued by replicating the payoffs using the original asset and a risk-free, zero-coupon bond.

An American Put Option

What is the value of a 1-year put option with exercise price 105 on a stock with current price 100?

The option can only be exercised now, in 6 months time, or at expiration.

= 31.5573% rF = 1% (per 6-month period)

Multi-Period Replication

Stock

100

80

125

100

156.25

64

Put (E=105)

5

0

41

P+

P-

Solving Backwards

125

100

156.25rF = 1%

5

0

P+

H = -0.089, B = -13.75 P+ = 2.64

80

64

100

41

5

P- rF = 1%

H = -1, B = -103.96 P- = 23.96 25!!-------

The put is worth more dead (exercised) than alive!

The Answer

100

80

125

25.00

2.64rF = 1%

H = -0.497, B = -64.11 P = 14.42

Assignments

• Reading– RWJ: Chapters 8.1, 8.4, 22.12, 23.2, 23.4– Problems: 22.11, 22.20, 22.23, 23.3, 23.4,

23.5

• Problem sets– Problem Set 1 due in 1 week